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Thermodynamics of microstructure evolution: grain growth
Victor L. Berdichevsky
Mechanical Engineering, Wayne State University, Detroit MI 48202 USA
(Dated: September 1, 2011)
1
Abstract
It is gradually getting clear that the macroscopic description of microstructure evolution requires
additional thermodynamic parameters, entropy of microstructure and temperature of microstruc-
ture. It was claimed that there is "one more law of thermodynamics": entropy of microstructure
must decay in isolated thermodynamic stable systems. Such behavior is opposite to that of ther-
modynamic entropy. This paper aims to illustrate the concept of microstructure entropy by one
example, the grain growth in polycrystals. The grain growth is treated within the framework of a
theory which is a modi�cation of Hillert theory. The modi�cation is made in order to reach simul-
taneously two goals: to get a coincidence of theoretical predictions with experimentally observed
results and to obtain the equations that admit analytical solutions. Due to these features, the mod-
i�ed theory is of independent interest. In the modi�ed Hillert theory one observes the decay of total
microstructure entropy when the system approaches the self-similar regime. The microstructure
entropy per one grain grows indicating a chaotization of grain sizes. It is shown also that there exits
an equation of state of grain boundary microstructure that links entropy of microstructure, energy
of microstructure, average grain size and a characteristic of the inhomogeneity of the large grain
distribution.
Keywords: microstructure entropy, con�gurational entropy, e¤ective temperature, grain growth
I. THERMODYNAMICS OF MICROSTRUCTURE EVOLUTION
By thermodynamics of microstructure one means a macroscopic description of bodies
possessing an evolving microstructure at a mesoscopic level. To describe a microstructure
evolution within the framework of classical thermodynamics, one has to specify a set of
macroscopic parameters characterizing the material and the microstructure, �1; :::; �k; and
the dependence of energy E on these parameters and on thermodynamic entropy S:
E = E (S; �1; :::; �k) : (1)
Then the governing evolution equations that respect thermodynamic laws follow from the
usual thermodynamic formalism. Applying this procedure to bodies with evolving microstruc-
tures, one faces the following problem: energy of the microstructure is not a function of
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macroscopic parameters of the microstructure and must be treated as an independent addi-
tional characteristic of the microstructure1. Consider why this occurs for the process of grain
growth in polycrystals.
If temperature of a polycrystal is increased to 0:2�0:3 of melting temperature, the grainsbegin to grow. The simplest macroscopic parameter that describes the state of the grain
boundary structure is the average grain size, R. It is introduced usually in the following way:
one measures the number of grains in the polycrystal, N , and its volume, jV j, and de�ne Ras
2R =
�jV jN
�1=3: (2)
Formula (2) corresponds to envisioning the average grain as a cube with the side 2R; if it is
a sphere of radius R (the usual choice), a numerical factor should be included in (2).
Suppose that grain boundaries are the only defects of crystal structure. Then energy of the
polycrystal is the sum of energy of heat motion of atoms, E0; and energy of grain boundaries,
Em; (subscript m stands for microstructure):
E = E0 (S) + Em: (3)
If Em were a certain function of R, then the usual thermodynamic formalism follows. This,
however, is not the case. Indeed, let all grain boundaries are high angle boundaries, and, thus,
the total energy of grain boundaries is the product of a constant, the grain boundary energy
per unit area, ; and the total area of grain boundaries. Then from dimension reasoning
energy of microstructure per unit volume, Um = Em /jV j ; can be written as
Um =EmjV j =
X
R; (4)
where X is a dimensionless parameter. It has the meaning of dimensionless boundary area,
X =grain boundary area � average grain size
total volume(5)
or dimensionless grain boundary energy,
X =grain boundary energy � average grain size
� total volume :
1 This point has been made in (Berdichevsky 2005, 2008).
3
FIG. 1: A grain boundary structure with dimensionless boundary area (6).
If X were a universal constant, then the evolution of grain boundaries can be treated within
the usual thermodynamic formalism. However, X does depend on microstructure. It is seen
if we �nd X for various microgeometries. For example, for tessellation of space in cubes of
equal sizes, X = 1:5, for Poisson-Voronoi tessellation (Ohser et al. 2000) X is about the
same, X = 1:45, but not precisely the same. Larger deviations one gets, for example, for
geometry shown in Fig. 1. The dimensionless boundary area X decreases as the ratio a2=a1
increases: as easy to see,
X = 1:51 + �
2
��1 + �3
2
�1=3; � =
a2a1: (6)
In the limit a2=a1 !1 X tends to 0.94. These examples show that X and R and, thus, Um
and R are independent. The independence of X and R can be seen also from consideration
of one grain: volume of any grain and area of its boundary are independent. One can �x the
grain boundary area and change the grain volume. The grain volume and grain boundary
area are linked only by the isoperimetric inequality. The same conclusion can be drawn
from studying the statistics of grain microstructures (Glicksman 2005, Glicksman 2005a,
Graner et al. 2000). So the evolving grain microstructure possesses at least two independent
macroscopic characteristics, Um and R.
Let us consider the simplest case when Um and R are the only characteristics needed in a
macroscopic theory, i.e. grain boundary structure is a two-parametric system (we ignore for a
moment thermodynamic entropy or temperature assuming that the grain growth occurs either
at adiabatic or at isothermal conditions). This case does not �t the usual thermodynamic
4
scheme because Um is not a certain function of R. However, we can set an analogy with
usual thermodynamic systems. The simplest example of thermodynamic system with two
independent parameters is an ideal gas. The ideal gas is characterized by temperature T and
mass density �: In this case the physical properties of the gas are described by the dependence
of free energy on T and �: One can choose energy per unit mass, U , and � as the independent
thermodynamic parameters of the gas. For such a choice, the physical properties of the gas
are described by the equation of state
S = S(U; �) (7)
where S is the entropy of gas per unit mass. Thermodynamics suggests the way for inde-
pendent measurements of S, U and �, and equation (7) admits an experimental veri�cation.
Remarkably, (7) holds for all su¢ ciently slow processes in gases. The grain boundary mi-
crostructure in polycrystals is, to some extend, similar: this microstructure is characterized
by a geometric parameter R and an energy parameter Um: Similar to (7) one can expect that
for grain boundary structures there might exist some "entropy", Sm; such that the state of
the grain boundary is described by an equation of state
Sm = Sm(Um; R); (8)
and for all processes the equation of state (8) holds true. The search of such equation of
state and the study of the accompanying issues is the subject of this paper. The starting
point for this study is Hillert theory of grain growth which provides an evolution equation for
probability density of grain sizes. As is known, Hillert theory does not yield the experimen-
tally observed probability distributions, and we begin with a modi�cation of Hillert theory
that provides proper probability density in the self-similar regime. Remarkably, the modi�ed
theory admits an analytical treatment. The modi�ed theory does support the existence of
the equation of state if the system is not far from the self-similar grain growth regime. We
de�ne entropy of microstructure, Sm; in such a way that it can be experimentally measured,
and the existence of equation of state (8) becomes an experimentally veri�able issue. As
could be expected from the very beginning, a hope for the existence of equation of state (8)
is overoptimistic, and there might be some additional parameters that a¤ect thermodynamics
of grain structure evolution. This turns out to be the case indeed: entropy of microstructure,
5
in addition to arguments mentioned in (8), depends on a characteristic of inhomogeneity of
the largest grains.
We show that, in accord with the previous conjecture (Berdichevsky 2005, 2008, 2009),
the total entropy of microstructure, Sm = jV jSm; decays indeed near the attractor, which inthe case under consideration, is the self-similar grain growth,
dSmdt
= jV j dSmdt
6 0: (9)
The modi�ed theory of grain growth yields, in addition, one more quanti�cation of the entropy
behavior: in unbounded polycrystals entropy of microstructure per one grain,
S� =SmN;
grows, indicating a chaotization of the grain structure. The decay of total entropy means
that in the product NS� the number of grains decays faster than S� grows.
The fact that the macroscopic description of microstructure evolution requires additional
thermodynamics parameters, entropy of microstructure and temperature of microstructure,
is being slowly surfaced up. These notions appeared under di¤erent names in theory of
amorphous materials (Cugliandolo et al. 1997, Sollich et al. 1997, Nieuwenhuizen 1997,
Nieuwenhuizen 1998, Sciortino et al. 1999, Barrat et al.2000, Nieuwenhuizen 2000, Sciortino
et al. 2001, Berthier et al. 2002, Berthier et al. 2002a, Ono et al. 2002, O�Hern et al.
2004, Haxton et al. 2007, Langer et al. 2007, Langer 2008, Bouchbinder et al. 2009),
granular materials (Edwards 1991, Edwards 1994, Makse et al. 2002, Potiguar et al. 2006),
plasticity theory (Berdichevsky 2005, 2006, 2008, Langer et al. 2010) and theory of composite
materials (Berdichevsky 2008, 2009). The most advanced implication of these notions to
phenomenology of the stress-strain curves in plasticity theory has been developed recently
by Langer, Bouchbinder and Lookman (2010). In (Cugliandolo et al. 1997, Sollich et al.
1997, Nieuwenhuizen 1997, Nieuwenhuizen 1998, Sciortino et al. 1999, Barrat et al. 2000,
Nieuwenhuizen 2000, Sciortino et al. 2001, Berthier et al. 2002, Berthier et al. 2002a, Ono
et al. 2002, O�Hern et al. 2004, Haxton et al. 2007, Langer et al. 2007, Langer 2008,
Bouchbinder et al. 2009, Langer et al. 2010, Makse et al. 2002, Potiguar et al. 2006) these
new parameters are called con�guration entropy and e¤ective temperature, in (Berdichevsky
2006, 2008, 2009) entropy of microstructure and temperature of microstructure. The term
6
entropy of microstructure seems preferable as it emphasizes the di¤erence from con�guration
entropy used in classical statistical mechanics for ergodic Hamiltonian systems. In contrast to
usual con�gurational entropy, entropy of microstructure corresponds to non-ergodic degrees
of freedom which fell out of the "ergodic bath". As was conjectured in (Berdichevsky 2005,
2005, 2008) and most explicitly mentioned in (Berdichevsky 2009, Sec. 2.7) there is "one
more law of thermodynamics"2: in isolated thermodynamically stable systems entropy of
microstructure decays. The origin of the decay is simple: the microstructure evolution is
described by a dynamic system in some phase space; the dynamic system is dissipative, and,
thus, possesses an attractor; the phase volume shrinks when the system is su¢ ciently close to
the attractor; accordingly, entropy of microstructure, which is associated with the logarithm
of the phase volume, decays. This picture assumes that the number of "dissipative" degrees of
freedom does not change. In fact, the situation is more complex. First, in order to introduce
a relevant phase space, one should distinguish leading and slave degrees of freedom, and this
is a non-trivial task. Second, the number of leading degrees of freedom usually decreases in
an isolated system. Due to that one can hardly give a more precise general statement rather
than a vague proposition on the decay of microstructure entropy of thermodynamically stable
systems near attractors in a generic case. Studying particular cases, like the case of grain
growth, becomes necessary to reveal the underlying physical mechanisms of entropy decay.
We begin with the classical in�nite-dimensional model of grain growth (Section 2) and a
detailed consideration of its �nite-dimensional truncation, which is, in essence, due to Hillert
(1965) (Section 3). This is a dissipative system in the phase space of grain sizes. To obtain
the dynamical system, which yields the experimentally observed distribution in the self-
similar regime, we consider dynamics of grain volumes. Grain volumes have the exponential
distribution in the self-similar regime (Berdichevsky 2011). This fact determines grain volume
dynamics almost uniquely (Section 4). Remarkably, the equation for probability density of
grain volumes can be put in the form of an equation studied analytically by Carr and Penrose
(1999). The Carr-Penrose analysis is reproduced here with some additional details (Sections
5, 6, 7). Entropy evolution is considered in Section 8, and the existence of the equation of
state in Section 9. The grain volume dynamics is meaningful for unbounded polycrystals,
2 I am thankful to A. Gorban for useful comments on the matter.
7
but in �nite ones it is intrinsically inconsistent (as Hillert grain size dynamics is). A way to
remove this inconsistency is considered in Appendix B. The range of change of microstructure
energy for di¤erent grain volume distributions is found in Appendix A.
II. GRAIN BOUNDARY DYNAMICS
Grain boundary being a surface of an arbitrary shape has an in�nite number of degrees
of freedom. Dynamics of this in�nite-dimensional system is usually assumed to by controlled
by the surface curvature,
vn = 2M H (10)
where vn is the normal velocity of grain boundary, M grain boundary mobility, the surface
energy density, H the mean curvature. Equation (10) must be complemented by the dynamic
equations for the lines at which grain boundaries meet, edges, and for the vertices, the points,
where the edges meet.
The mechanism of grain growth has been revealed by von Neumann (1952). In two-
dimensional case there is an exact relation (Neumann 1952, Mullins 1956) between the rate
of the area of a grain, A, and the number of its neighbors, �:
dA
dt= const (� � 6) : (11)
Therefore, the grains with the number of neighbors larger than six grow, while the grains
with smaller number of neighbors disappear. Relation (11) has been generalized to three-
dimensional structures by Cahn (1967) and MacPherson and Srolovitz (2007).
Dynamics of grain boundaries is extremely complex, and in order to get some insight one
has to make a �nite-dimensional truncation. To develop a truncation we will use energy
approach. Therefore, we need �rst to see how to obtain an "exact" equation (10) from the
standard energy reasoning of continuum mechanics.
In polycrystals, grains grow to reduce energy of grain boundaries. There are several other
factors involved such as interaction of grain boundaries with impurities, dislocations and
dislocation cells, inhomogeneity of internal stresses, etc.; we ignore all these factors to obtain
as simple picture of grain growth as possible. Total energy of grain boundary, E , is an integral
8
of surface energy density, ; over the grain boundaries :
E =Z
dA; (12)
dA being the area element on : Surface is the union of all grain boundaries of the
polycrystal. We assume that misorientation of neighboring grains is large, and, therefore, is
a material constant. We assume also that the grain growth occurs at a constant temperature
and identify E with total energy of the system, thus omitting energy of heat motion ofmolecules that plays the role of an additive constant.
Motion of grain boundaries causes a dissipation, D; that is equal to the rate of energy
decrease.dEdt= �D: (13)
The dissipation that corresponds to equation (10) is
D =
Z
1
Mv2ndA: (14)
To obtain dynamic equations for grain boundary evolution, one needs a variational equation,
which holds for arbitrary variations of current grain boundary position, �r: This equation
must transform to energy equation, if �r is replaced by the actual change of grain boundary
position, v�t: This variational equation is
�E =Z
1
Mvn�rndA; (15)
where �E is the variation of energy due to variation of grain boundary position, and �rn thevariation of the grain boundary position along the normal to the boundary.
It is convenient to write (15) in terms of the so-called dissipative potential, D. For thequadratic functional of velocities (14), the dissipative potential di¤ers from dissipation by
the factor 12:
D =12D: (16)
The variational equation can be written as (see Berdichevsky 2009, Sect. 13.3)
�E = ���D: (17)
9
Here the operation ��corresponds to variation of the functional over velocity and subsequent
change of variation of velocity by �r :
��D =
Z
1
Mvn�rndA: (18)
For actual variation of grain boundaries the variational equations (15) and (17) transform to
the energy equation (13). If motion of an edge causes some dissipation, then velocity of the
edges enter (14); additional energy associated with edges will modify (12) as well. Some of
the corresponding e¤ects have been discussed by Gottstein and Shvindleman (1999).
Grain growth is one of the most advanced Chapters of material science. It has been studied
intensively during last century, and the literature on the subject is extremely rich. Note here
the reviews by Atkinson (1988), Cahn (1996), Humphreys and Hatherley (2004), Gottstein
and Shvindlerman (1999), the pioneering papers by Smith (1952), Feltham (1957), and Hillert
(1965), a short fundamental note by von Neumann (1952), followed by generalizations by
Mullins (1956), Cahn (1996) and MacPherson ans Srolovitz (2007), the papers by Pande
(1987) and Pande et al. (2010), Jeppson et al. (2008), Glicksman (2005, 2005a), Rios et al.
(2008), Vandermeer et al. (1994), numerical simulations by Anderson et al. (1984), Rollet et
al. (1989), Srolovitz et al. (1984), Wakai et al. (2000), Wang et al. (2009), the experimental
studies by Rhines et al. (1974), Marsh et al. (1993), Zhang et al. (2004), Vandermeer et al.
(2003), Rowenhorst et al. (2010).
III. GRAIN SIZE DYNAMICS
To study the dynamic system described in the previous Section one has to make a �nite-
dimensional truncation. An attractive idea is to characterize each grain by a small number of
parameters and develop a system of dynamical equations for these parameters. The simplest
option is to characterize each grain by just one number, the grain size. The grain size is
usually introduced as the radius of the sphere with the volume equal to the volume of the
grain. So, the ensemble of N grains is characterized by N numbers, r1; :::; rN : To derive
the dynamic system for r1(t); :::; rN(t) we have to specify the dependence of energy and
dissipation on r1(t); :::; rN(t): We accept two approximate relations: energy of a grain of size
r is 4�r2 ; and dissipation caused by the change of grain size is 4�r2 _r2=M ( _r � dr(t)=dt):
10
Accordingly, total energy and total dissipation are
E = 1
2
NXa=1
4�r2a ; D =1
2
NXa=1
4�r2a _r2a=M: (19)
The factors 12caused by counting grain boundaries twice in sums over all grains (19).
The dynamic equation for r1(t); :::; rN(t) follow from the variational equation (15). The
essential additional condition is that variations �r1(t); :::; �rN(t) are not arbitrary: the total
volume of the grains must be equal to the volume of polycrystal jV j ;
NXa=1
4�
3r3a = jV j : (20)
Besides, of course, there are obvious constraints
ra > 0; a = 1; :::; N: (21)
Introducing the Lagrange multiplier � for the constraint (20), we obtain from (19), (15) for
ra > 0
4�ra � �4�r2a = �4�r2a _ra=M; a = 1 ; :::; N
or, rearranging terms and replacing � by � ;
dradt=M
�� (t)� 1
ra
�; a = 1; :::; N: (22)
In what follows, N means the number of grains with positive volume. This number changes
in time. When ra(t) reaches zero, we set
dradt= 0 for ra = 0: (23)
The Lagrange multiplier � (t) is determined by the condition (20). This condition allows
one to �nd � as an explicit function of ra (t) : multiplying (22) by r2a and summing over a we
get
� (t) =NXa=1
ra (t)
,NXa=1
r2a(t): (24)
Remarkably, this simple dynamic system captures qualitatively the mechanism of grain
growth mentioned in the previous Section: as follows from (11), there is a critical number
of grain sides (6 in 2D), and grains with the number of grain sides larger than critical grow,
11
while the grains with the number of grain size smaller than critical decay. Experiments show
that the number of grain sides is proportional to the grain size. Therefore, su¢ ciently large
grains grow while smaller grains decay. This is precisely the phenomenon that the system
(22) describes. The solution of the system behaves in the following way: the grains with
the size r (t) larger than 1=� grow, while the grains with the size smaller than 1=� shrink
and vanish. The number of grains, N , decreases until it becomes equal to unity: one grain
survives and occupy the entire volume. At this point the process stops.
The system (22)-(24) has the singularity at ra = 0 : the smaller the grain the faster it
is vanishing. This singularity can be regularized in various ways to capture the vanishing
process more realistically, but we do not dwell on this issue here focusing mostly on grain
size evolution of non-small grains.
A qualitative picture of grain dynamics within the model (22)-(24) is relatively simple.
Consider �rst a polycrystal containing only two grains. In this case, the phase space is
two-dimensional. Admissible values of the grain sizes lie on the curve (see Fig. 2)
r31 + r32 = R3;4�
3R3 = jV j :
The system (22) has the only equilibrium point r1 = r2 = R=21=3; i.e. if the grain sizes are
equal, no further evolution occurs. The equilibrium point is unstable: any di¤erence in grain
sizes yields growing of the larger grain and vanishing of the smaller one. The attractor of the
system consists of two points, (R; 0) and (0; R). The basins of the attracting points are the
parts of the curve on the right and on the left of the equilibrium point.
Consideration of three grain system brings one more element in this picture. The phase
�ow of three grain system is shown in Fig. 3. The system has again the only equilibrium
point r1 = r2 = r3 = R=31=3; which is unstable. The two-dimensional surface of admissible
grain sizes,
r31 + r32 + r33 = R3;
is split into three basins of attracting points (R; 0; 0) and (0; R; 0) and (0; 0; R):What we see
additionally in this system is the process of decrease of the number of grains: for a randomly
chosen initial position the phase trajectory �rst slides into two-dimensional phase space, and
then goes to the attracting point (a phase trajectory is shown in Fig. 3 by the bold curve).
12
FIG. 2: Phase �ow for two grains.
FIG. 3: Phase �ow for three grains.
Such successive sliding into phase spaces of smaller dimensions remains typical for ensembles
of large numbers of grains.
Our goal is to describe the evolution of a very large number of grains. Remarkably, for
large N the dynamic system (22)-(24) can be replaced by a continuum. This is caused by
13
the following feature if this system: if r1(t0) < r2(t0) at some instant t0; then at all times
r1(t) < r2(t)
until both grains vanish and r1 = r2 = 0: Indeed, from (22)
d (r2 � r1)
dt=dr2dt� dr1
dt=M
�1
r1� 1
r2
�=M
r2 � r1r1r2
: (25)
Hence, if r2 � r1 > 0 initially, the derivative of r2 � r1 is positive, and r2 � r1 can only grow.Let at t = 0 there be N0 grains, and the initial sizes of the grains be�r1; :::;�rN0 :We assume
that the grains have di¤erent sizes and arrange the sizes in the decreasing order,
�r1 >�r2 > :::: >�rN0 :
The maximum initial grain size, �r1 � �rmax; is a number on a segment
[0; R]�R = (3 jV j =4�)1=3
�: Let us introduce a one-dimensional continuum with Lagrangian
coordinate �r and Eulerian coordinate r, r = r(t; �r); the velocity of which is de�ned by the
equation,@r(t;�r)
@t=M
�� (t)� 1
r(t;�r)
�: (26)
Here � (t) is some not-yet-de�ned function of time. The "mass density" of such continuum,
�(t; r); obeys the equation
@�
@t+
@
@rM
�� (t)� 1
r
�� = 0: (27)
If we set
� (0; r) =NXa=1
� (r ��ra) (28)
and
� (t) =
RZ0
� (t; r) rdr
, RZ0
� (t; r) r2dr; (29)
then the solution of (27) is
� (t; r) =
NXa=1
� (r � ra(t)) ; (30)
where ra(t) is the solution of (22). Indeed, let �(t; r) be the function (30). Consider motion of
the continuum (26). The points with initial conditions r(0;�r1); :::; r(0;�rN)move in accordance
with the equations (22), because for �(t; r) given by (30), functions (29) and (22) coincide.
14
The general solution of (27) is
�(t; r)@r(t;�r)
@�r= �(0;�r):
Denote by�r(t; r) the inversion of function r(t;�r) with respect to�r: For any c and the relation
holds
� (r(t;�r)� c)@r
@�r= � (�r ��r(t; c)) :
It can be checked by multiplying by a trial smooth function ' (�r) and integrating over �r:
Hence,NXa=1
� (r � ra (t))@r(t;�r)
@�r=
NXa=1
� (�r ��ra (t; ra)) =NXa=1
� (�r ��ra) ;
i.e. �(t; r) is a solution of (27) with the initial condition (28).
Now we can consider solutions of (27) for smooth positive initial data
� (0; r) = �0 (r) ; 0 6 r 6 R; (31)
identifying the quantity � (r)�r with the number of grains in the interval [r; r +�r] for small
�r: Obviously, there must not be "grain size �ow" at r = R. From (29) � > 1=R; therefore
vanishing of the grain size �ow M (�� 1=R) � at r = R yields
� (t; R) = 0: (32)
The initial boundary-value problem (31), (32) for equation (27) is nonlinear due to the link
between � (t) and � (t; r) (29).
Note that formula for � (29) can be replaced by the condition that the total volume of
the grains coincide with the prescribed polycrystal volume,
RZ0
4�
3r3� (t; r) dr =
4�
3R3: (33)
Then (29) follows from (27), (32) and (33) by integrating (27), multiplied by r3; over r.
The integral,RZ0
� (t; r) dr;
15
has the meaning of the total number of grains at the instant t, N(t). Integrating (27) over r
we get the evolution law for N(t) :
dN
dt= M
�� (t)� 1
r
��
����r=0
: (34)
Here we used condition (32). Since the right-hand side of (34) must be �nite, � � r as r ! 0;
and (34) simpli�es todN
dt= �M lim
r!0
�(t; r)
r: (35)
Equation of the type (27), (29) appeared for the �rst time in the work by Todes and
Khrushchev (1947); formulas for velocity and � were di¤erent and corresponded to the evolu-
tion of a set of precipitates in the mother matrix phase (so-called Ostwald�s ripening). Lifshitz
and Slyosov (1958) developed a slightly di¤erent equation in the same physical context. Their
essential new step was the integration of this equation in the case of self-similar growth of par-
ticles (see also Lifshitz, Slyosov (1961), Lifshitz, Pitaevskii (1981)). Wagner (1961) applied
Lifshitz-Slyosov arguments to study an equation which transforms into Todes-Khrushchev
equation and Lifshitz-Slyosov equation in particular cases. Lifshitz and Slyosov (1958, 1961)
argued that there is a unique self-similar distribution of particle sizes toward which every ini-
tial distribution must converge. This was questioned by Brown (1989, 1992), who suggested
that there are many self-similar distributions; each one is selected by the initial distribution
of the largest particles. The situation was cleared up considerably in mathematical studies
by Carr and Penrose (1998), Carr (2006), Niethammer and Pego (1999) and Niethammer
and Velázquez (2006). These studies supported the Brown proposition.
Equation (27) was �rst applied to studying of grain growth by Hillert (1965). He used
di¤erent argumentation in derivation of (27). Hillert theory is called often a mean �eld theory
of grain growth, because the interaction of grains, described by � (t), is the same for all grains.
Note, however, that, as follows from the derivation given here, there is an essential di¤erence
from the usual mean �eld theories, considering just one particle in an e¤ective �eld: the
above-described theory is "exact" within the framework of the assumptions for energy and
dissipation (19). In this sense, it is not a "mean �eld theory", but a "precise truncation" of
the in�nite-dimensional system of Section 2.
The self-similar grain size distributions following from equation (27) deviates from ex-
perimental data. Therefore, before getting down to the entropy behavior in grain growth
16
we modify equation (27). Another goal of such modi�cation is to make it accessible for an
analytic treatment.
IV. GRAIN VOLUME DYNAMICS: SETTING OF THE PROBLEM
Phase �ow in grain volume phase space. It was shown in (Berdichevsky 2011) that the
most chaotic microgeometry of polycrystals corresponds to exponential distribution of grain
volumes, and that the exponential distribution �ts well to experimental data for the self-
similar grain growth. Therefore, it is interesting to develop dynamical equations in phase
space of grain volumes that result in exponential distribution for the self-similar evolution.
In this Section such a system of dynamic equations for grain volumes is derived.
Denote by � (t; v) the density of grain volumes: � (t; v)�v is the number of grains with
volumes lying in a small interval [v; v +�v] : Grain volume density � (t; v) obeys the equation
@�
@t+
@
@v(V (t; v) �) = 0; (36)
where V (t; v) is the velocity of the phase space �ow.The number of grains,
N =
jV jZ0
� (t; v) dv; (37)
changes in time, and so does the average grain volume,
�v(t) =jV jN(t)
: (38)
We assume that the explicit dependence of V (t; v) on time can occur only through thedependence of V (t; v) on �v3:
V = V (�v(t); v) :
Besides, we accept that the only physical constants that a¤ect rate of the process are the
grain boundary mobility M and grain boundary energy : Function V(�v; v;M; ) has the
3 This feature is similar to that of Hillert theory, because �v is expressed in terms of � (t; v) from (37), (38):
in Hillert theory, as follows from (27), (29), the phase velocity depends explicitly on time only through
� (t; r) :
17
dimension volume/time. From dimension theory, such a dependence must be of the form,
V(t; �v) =M �v�1=3�(u); u =v
�v;
u being the relative grain volume, � (u) dimensionless function of u.
Probability distribution of grain volumes, f(t; v), is de�ned as
f(t; v) =1
N(t)� (t; v) =
� (t; v)jV jR0
� (t; v) dv
: (39)
The self-similar evolution corresponds to a stationary distribution of relative grain volumes,
g(u), u = v=�v (t) :
f(t; v) =1
�v(t)g
�v
�v(t)
�: (40)
The exponential function g(u) = e�u is a solution of equation (36) only4 if �(u) is a
linear function of u. Hence, V is a linear function of v. This is the major motivation fora modi�cation of Hillert theory. In this Section we are going to give some physical reasons
that yield a linear dependence of V on v. After an appropriate time scaling, this dependencetakes the form
V(t; v) = 1
�v2=3(t)(v � �v(t)) : (41)
The corresponding equation for grain volume density is
@�
@t+
@
@v
�1
�v2=3(v � �v) �
�= 0: (42)
The physical reasons for (41), (42) are as follows.
Energy and dissipation. Consider a set of N grains with volumes v1; :::; vN : Grain volumes
evolve in such a way that the total volume is conserved:
v1 + :::+ vN = jV j : (43)
To derive dynamic equations for v1(t); :::; vN(t); we have to specify energy and dissipation.
Let us �rst write formulas (19) in terms of grain volumes:
E = 1
2
NXa=1
4�
�va4�=3
�2=3; D =
NXa=1
4�
M
�va4�=3
��2=3( _va)
2 : (44)
4 Besides, �v�1=3d�v=dt must be constant.
18
These energy and dissipation yield Hillert dynamics of grain sizes just written in terms of
grain volumes. Let us rewrite (44) referring each term to its value for va = �v :
E = 2� �
�v
4�=3
�2=3 NXa=1
�va�v
�2=3; D =
4�
M
��v
4�=3
��2=3 NXa=1
�va�v
��2=3( _va)
2 : (45)
We will use a di¤erent approximation aiming to get for the phase velocity V(t; v) a linearfunction of v. We set
E = 2� �
�v
4�=3
�2=3 NXa=1
'�va�v
�; D =
4�
M
��v
4�=3
��2=3 NXa=1
�va�v
�_v2a; (46)
where ' and are some functions to be found. Function ' has the meaning of grain energy
normalized by grain energy of an "averaged grain". From comparison (45) and (46) we see
that in Hillert theory the normalized grain energy is
'(u) = u2=3; (47)
while the dissipation factor, ; depends on the relative grain volume as
(u) = u�2=3: (48)
Emphasize that the formulas of Hillert theory (45) (and, thus, (47) and (48)) are highly
approximate: they are based on equating the surface of a grain to the grain volume in the
power 2=3 and including a factor chosen in such a way that the equality holds for spherical
grains. Since no grain is spherical, one can only hope that such an approximation holds on
average for a large ensemble of grain. One can try other functions instead of (47), (48), as
we will do.
Let us plug (46) into variational equation (17) and assume that during some short interval
none of grains disappears, i.e. N(t) = const and �v(t) = const: We obtain the dynamic
equations holding during that period of time:
2�
��v
4�=3
�2=31
�v'0�va�v
�� �(t) = �4�
M
��v
4�=3
��2=3 �va�v
�_va: (49)
Here �(t) is the Lagrange multiplier for the constraint (43). We seek functions ' and , for
which _va can be linear functions of va. This occurs only if 1= and '0= are linear functions
of u. Setting
= (a0u+ u0)�1; '0= = a1u+ u1; (50)
19
FIG. 4: Dependence of normalized grain energy ' in Hillert theory and modi�ed theory on relative
grain volume.
a0; u0; a1; u1 being constants, we obtain for relative energy:
' (u) =a1a0u+
u1 � u0a1=a0a0
ln (a0u+ u0) + c: (51)
So our model contains 5 parameters, a0; a1; c; u0; u1:
Experimentally measured values of u are usually in the range 0 6 u 6 6: One can choosethe constants a0, u0; a1 and c in such a way that functions ' (u) and (u) approximate the
corresponding functions of Hillert theory, u2=3 and u�2=3, reasonably well for 0 6 u 6 6: Forexample, setting
' (0) = 0; ' (1) = 1; '0 (1) = 2=3; ' (5) = 52=3; (1) = 1;
one gets
u0 = 0:2; u1 = 0:4; a0 = 0:79; a1=a0 = 0:33; c = 0:67: (52)
These are the values that will be used further for numerical estimates. The graphs of relative
grain energy (51) and that of Hillert theory are shown in Fig. 4. It is seen that they are
practically coincided in the experimentally observable range and di¤er only for extremely
large relative grain volumes; energy of Hillert theory grows faster. Comparison of function
(u) (50) with that of Hillert theory is shown in Fig. 5. Considerable di¤erence is observed
20
FIG. 5: Dependence of the dissipation factor on relative grain volume in Hillert theory and
modi�ed theory.
only for very small grains: in the modi�ed theory, disappearance of small grains occurs with
much smaller dissipation (resistance).
There is an essential distinction of the modi�ed theory from Hillert theory: energy of a
grain in Hillert theory depends only on the grain volume, while in (46) grain energy depends
also on the average grain size. The latter accounts for the dependence of grain energy on
the "environment" for large grains (as is seen from Fig. 4, for relative volumes u 6 6 bothenergies are practically identical). As to dissipation, a choice for parameters corresponding
to (1) = 1; makes dissipations identical for the grains, volume of which is equal to the
average grain volume.
Consider now the case when �v changes. Change of �v does not alter equations (49). Indeed,
the dependence of energy on �v bring one more term into energy equation,
@E@�v
d�v
dt= �@E
@�v
�v
N
dN
dt=
@E@N
dN
dt: (53)
Ifm grains disappeared during some small time �t; then the energy �ux (53) must be balanced
with energy of the disappeared grains
mXk=1
@E@va
�����va=0
dvadt
����va=0
: (54)
21
Here dva=dt can be taken from (49). Energy �ux can be treated as a dissipation. No changes
occur in dynamical equations for grains with nonzero volumes.
Grain volume dynamics. So, the dynamic equations take the form
1
M_va = �(t)
1
4�
��v
4�=3
�2=3 �ava�v+ u0
��
2
��v
4�=3
�4=31
�v
�bva�v+ u1
�: (55)
Time t in (55) is physical time, denote it by tphys: It is convenient to introduce a scaled time
t =M
2 (4�=3)1=3u1a� bu0a+ u0
tphys: (56)
It has the dimension length2: De�ning a new variables �1; �! �1;
� = 2� 1
(4�=3)2=3 �v1=3�1;
we put (55) in the form
dvadt
= �v1=3 (�1a� b)a+ u0
u1a� bu0
�va�v� u1 � �1u0
�1a� b
�: (57)
It follows from conservation of volume (43) that �1 is a constant determined from the equation
u1 � �1u0�1a� b
= 1;
from which
�1 =u1 + b
a+ u0: (58)
Plugging (58) into (57) we put the dynamic equations into the �nal form:
dvadt
=1
�v2=3(va � �v) ; va > 0: (59)
It must be complemented the condition
dvadt
= 0 if va = 0: (60)
Function �v(t) is determined by the volume conservation condition (43), in which N(t) is the
number of non-zero functions va(t): From (59) and (43),
�v(t) =v1(t) + :::+ vN(t)
N(t): (61)
22
Grain volume density. Grain volume evolution is monotonous as grain size evolution is:
if v1 is smaller than v2 initially, v1 remains smaller v2 for all times until v1 = v2 = 0: This
allows us to introduce a continuum, v = v(t;�v); with the velocity �eld
@v(t;�v)
@t=
1
�v2=3(v (t;�v)� �v) : (62)
The equation for the grain volume density, �(t; v); takes the form (42).
Equation (42) is to be complemented by the initial and boundary conditions,
� (0; v) =�� (v) 0 6 v 6 jV j ; � (t; jV j) = 0: (63)
Function �v(t) is linked to � (t; v) by the condition of conservation of total volume:
jV jZ0
v� (t; v) dv = jV j : (64)
This closure makes the problem non-linear.
Function �v(t) can be expressed in terms of � (t; v) explicitly. To this end one multiplies
(42) by v and integrate over v on [0; jV j] : Then
�v(t) =
jV jR0
v� (t; v) dv
jV jR0
� (t; v) dv
: (65)
The same relation follows, of course, from (61), if we note that the number of grains is
N(t) =
jV jZ0
�dv; (66)
while
v1 + :::+ vN =
jV jZ0
� (t; v) vdv:
Note that in (42), in contrast to (27), velocity �eld is not singular at v = 0, and � (t; v)
is allowed to have a �nite value at this point. Note also that, as in �uid mechanics, "mass"
� (t; v) may be discontinuous on the lines in (t; v)�space that are trajectories of the particlesof the �ow (62).
23
The mathematical problem for � can be transformed to the problem studied by Carr and
Penrose (1998). To this end we consider function
�1 (t; v) =1
jV j� (t; v) :
It satis�es the equation,@�1@t
+ �v1=3@
@v
�v�v� 1��1 = 0;
where
�v =
jV jZ0
v�1dv
, jV jZ0
�1dv: (67)
Besides, for �1 the normalization condition holds:
jV jZ0
v�1dv = 1: (68)
Due to (68), equation (67) can be written also as
1
�v=
jV jZ0
�1dv: (69)
Changing additionally time variable, t! t1;
�v1=3dt = dt1;
we arrive at the equation@�1@t1
+@
@v
�v�v� 1��1 = 0;
with the constraint (68) and a link between �v and �1 (69). It has been studied by Carr and
Penrose (1999) in in�nite domain, jV j = 1: They showed that this equation can be solved
analytically.
Since formulas following from (42) have some physical meaning, and there are details of
physical signi�cance absent in the Carr-Penrose treatment we repeat their analysis in case of
equation (42). Besides, we consider also the grain volume dynamics for polycrystals of �nite
volume (Appendix B).
24
In conclusion of this Section note that the grain volume probability density f(t; v) de�ned
by (39) obeys the equation that does not have a divergence form:
@f
@t+
@
@v
1
�v2=3(v � �v) f = f
1
�v
d�v
dt: (70)
Equation (70) follows from (42) and conservation of volume, N�v = jV j ; which yields,
1
N
dN
dt= �1
�v
d�v
dt: (71)
The o¤-divergence term is caused by the change of the number of grains in the evolving grain
structure.
V. SELF-SIMILAR EVOLUTION
From (39) and (40), in the self-similar evolution the grain volume density has the form,
� (t; v) =N(t)
�v(t)g
�v
�v(t)
�=jV j�v(t)2
g
�v
�v(t)
�: (72)
Plugging (72) in (42), we obtain equation for g(u):
(u (1� q)� 1) dgdu= (2q � 1) g (73)
where we introduced the notation
q =1
�v1=3d�v
dt: (74)
Obviously, in (73) q must be a constant. This determines the rate of change of the average
grain volume
�v2=3(t) = �v2=3(0) +2
3qt: (75)
Note that q cannot be negative. Indeed, integrating (42) over v we get
dN
dt= �
�1
�v2=3(v � �v) �
�jV j0
= ��v1=3� (t; 0) : (76)
Since � > 0; N decreases and stops changing only when � (t; 0) reaches zero. Accordingly, it
follows from N�v = jV j ; that��v > 0:
The average grain size is proportional to �v1=3 and, thus, changes as t1=2 in the self-similar
regime.
25
There are �ve distinct cases, q = 1, q > 1, 1=2 < q < 1, q = 1=2, 0 6 q < 1=2: We have
q = 1 : g(u) = const e�u (77)
q > 1 : g(u) =const
(1 + (q � 1)u)2q�1q�1
(78)
1=2 < q < 1 : g(u) = const
����u� 1
1� q
���� 2q�11�q
(79)
q = 1=2 : g(u) = const (80)
0 6 q < 1=2 : g(u) = const
,����u� 1
1� q
���� 1�2q1�q
: (81)
If the polycrystal is unbounded (0 6 u 61) and g(u) is non-zero on [0;1] ; the convergenceof integral of g leaves only the possibility q > 1: Since function g(u) may have �nite support,one can not exclude the solutions (79)-(81). Besides, g(u) may have jumps. Therefore, for
example, a function
g(u) =
8>>><>>>:const e�u 0 6 u 6 a
0 a 6 u 6 b
const e�u b 6 u <1
is also a solution.
Further three solutions, (77), (78) and (79), will play a special role. The latter one will
be used only on the segment [0; 1= (1� q)] ; we set
g(u) =
8<: const�
11�q � u
� 2q�11�q
0 6 u 6 11�q
0 11�q 6 u <1
: (82)
As follows from (89), (77) and (78), for an unbounded polycrystal g(u) must obey the
normalizing conditions,1Z0
g(u)du = 1;
1Z0
g(u)udu = 1: (83)
These conditions specify the arbitrary constants in (77), (78) and (82) uniquely. It is con-
venient to write the �nal result in terms of the power, n = (2q � 1) = (1� q) ; instead of
parameter q. We have:
n =1 g(u) = e�u; 0 6 u 61 (84)
26
FIG. 6: Self-similar probability densities of relative grain volumes: exponential distribution (84)
and power distributions (85), (86) for n=8; points are the experimental data from Zhang et al.
(2004).
2 < n <1 g(u) =
8<: n+1(n+2)n+1
(n+ 2� u)n 0 6 u 6 n+ 2
0 n+ 2 6 u <1(85)
2 < n <1 g(u) =(n� 1) (n� 2)n�1
(n� 2 + u)n 0 6 u <1: (86)
Both distributions, (85) and (86) converge to the exponential distribution (84) as n tends
to in�nity. In experiments, the relative grain volume changes within the limits [0; 6] and
for u > 6 the probability density is practically zero. Three functions (84), (85) and (86) are
within the experimental errors for n & 8 (Fig. 6).If probability density has at t = 0 the forms (84) or (85), it remains unchanged inde�nitely
(this follows from further equations (89) and (94).
VI. GRAIN VOLUME DYNAMICS: EXACT SOLUTION
To study the convergence to self-similar solutions, we rewrite equation (42) in terms of
probability density of relative grain volumes, g(t; u);
g(t; u) = �vf(t; v) =�v
N� (t; v) = �v2(t)� (t; v) /jV j ; u � v/ �v(t): (87)
27
It is convenient to modify the time variable, t! � ; as
d�
dt= �v�2=3(t): (88)
Then
�v2=3@
@t=
@
@�;
and@g
@�+ (u (1� q (�))� 1) @g
@u= (2q (�)� 1) g; (89)
where q (�) is the function (74). According to (74) and (88), q (�) controls the evolution of
average grain volume. In terms of the modi�ed time � the parameter q (�) has the form
q (�) =d ln �v
d�or �v (�) = �v(0)e
R �0 q(�
0)d� 0 : (90)
Equation (89) is considered in the region � > 0; 0 6 u 6 u� (�) ; u� (�) = jV j /�v (�) : Itmust be complemented by the initial condition,
g (0; u) =�g (u) ; (91)
the boundary condition,
g (� ; u� (�)) = 0; (92)
and the normalization condition,
u�(�)Z0
g (� ; u) du = 1; (93)
which follows from the relation between � and g (87) and the conservation of total volume
jV j = N�v:
The normalization condition (93) yields the relation linking q (�) and g (� ; u):
q (�) = g (� ; 0) : (94)
It is obtained by integrating (89) over u and using (92) and (93).
Note the relation which follows from (87) and (65),
u�(�)Z0
ug (� ; u) du = 1: (95)
28
This relation is not an additional one to (89)-(93): integration of (89), multiplied by u; over
u and using (92), (93) yields
d
d�
u�(�)Z0
ug (� ; u) du+ 1 =
u�(�)Z0
ug (� ; u) du:
Therefore, (95) holds true as soon as it holds at the initial instant.
Equation (89) can be solved exactly. To this end we introduce the continuum u (� ;�u)
de�ned by the equation
@u (� ;�u)
@�= (1� q (�))u (� ;�u)� 1; u (0;�u) = �u: (96)
Here q (�) is viewed as a known function. Solution of (96) can be written explicitly. Denote
by m(�) and M(�) the functions determined from di¤erential equations
dM (�)
d�+ (1� q (�))M (�) = 0; M(0) = 1; (97)
dm (�)
d�=M (�) ; m(0) = 0:
Then, as can be checked by direct inspection,
M(�)u (� ;�u) +m (�) = �u: (98)
Changing the space variable in (89), u!�u; we put (89) in the form
@g (� ;�u)
@�= (2q (�)� 1) g (� ;�u) ;
which yields the solution
g (� ; u) =�g (�u) eR �0 (2q(�
0)�1)d� 0 ; �g (�u) � g (0;�u) :
Finally, returning to the original space variable, we have
g (� ; u) =�g (M (�)u+m (�)) eR �0 (2q(�
0)�1)d� 0 : (99)
To complete the solution, we have to link q (�) with �g (�u) : From (94) we get
q (�) =�g (m (�)) eR �0 (2q(�
0)�1)d� 0 : (100)
29
Equations (100) and (97) form a closed system of integral-di¤erential equations for q (�) ;
m (�) and M (�). Remarkably, solution of this system can be reduced to solution of an
algebraic equation. Let us de�ne two functions associated with the initial probability density,
G(u) =
Z u�
u
�g (u0) du0; H(u) =
Z u�
u
G (u0) du0; u� � u� (0) : (101)
Note that due to (93) and (95) and the de�nition of G and H
G(0) = 1; G(u�) = 0; H(0) = 1; H(u�) = 0: (102)
Since, from (97),dm
d�=M(�) = e
R �0 (q(�
0)�1)d� 0 ; (103)
equation (100) can be written as
d
d�e�
R �0 q(�
0)d� 0 =dG (m (�))
dm
dm
d�: (104)
This equation is integrated to
G (m (�)) = e�R �0 q(�
0)d� 0 : (105)
The constant of integration was determined by the initial conditions m (0) = 0 and G(0) = 1:
Replacing in (104) G by - H 0 and invoking again (103) we get the �nal equation for m (�)
which allows one to �nd m (�) as soon as initial grain volume distribution g(�u) is known,
H (m (�)) = e�� : (106)
Afterm (�) is found,M(�) and q (�)are determined from (97), while the current grain volume
distribution is given by (99). This must be complemented by the link between time t and
auxiliary time � following from (88) and (90):
t = �v(0)
Z �
0
e23
R ~�0 q(�
0)d� 0d~� : (107)
Function q(�) can be written explicitly in terms of initial data and m (�) : From (97),
q(�)� 1 = d
d�lndm
d�:
On the other hand, according to (103)-(105)
dm
d�=H (m (�))
G (m (�)): (108)
30
Hence,
q = 1 +d lnH
d�� d lnG
d�= �G
0 (m (�))
G (m (�))
dm
d�=�g (m (�))H (m (�))
G2 (m (�)): (109)
Note a convenient form of solution which employs only function m (�) :
g (� ; u) =�g
�dm (�)
d�u+m (�)
�dm
d�
�G (m (�)) : (110)
It follows from (99), (103) and (104). Another convenient form of the solution is
g (� ; u) = �g
�dm (�)
d�u+m (�)
�� 1Z0
�g
�dm (�)
d�~u+m (�)
�d~u: (111)
It follows from (111) and the de�nition ofG (101). Note also the relations between dm (�) =d� ;
G(m) and the average grain volume �v (�) ; which is a consequence of (90), (105), and (103)
dm
d�=�v (�)
�v (0)e�� ; G(m(�)) =
�v (0)
�v (�): (112)
VII. EVOLUTION TO SELF-SIMILAR REGIME IN UNBOUNDED POLYCRYS-
TAL
Let the polycrystal be unbounded, jV j = 1: Accordingly, u�(�) = jV j/ �v (�) = 1: The
behavior of the solution as � !1 depends essentially on the initial data. Carr and Penrose
(1999) showed that g (� ; u) can converge to steady solutions of Section 5, or it may not
converge at all and keep oscillating at large times. Small changes of initial data result in
small changes in the solution. Therefore, for practical purposes it is enough to properly
approximate initial data. In experiments �g(u) = g (0; u) has always a �nite support, i.e.
�g (u) is not zero for 0 6 u 6 a; and vanishes for u > a; a being a �nite number. It is widely
accepted that many physical properties of polycrystals, including plasticity and fracture, are
mostly a¤ected by the distribution of largest grains. Interestingly, the same is true for grain
growth. As will be seen further, the asymptotics of g (� ; u) at large � is determined by the
distribution of largest grains, i.e. by the behavior of �g (u) in a vicinity of the point u = a.
Therefore, an approximation of this behavior plays a special role. We will use for �g (u) in
some vicinity [a0; a] of point a the function,
�g (u) =
8<: c0 (a� u)n (1 + c1 (a� u)) a0 < u 6 a
0 u > a: (113)
31
The size of the vicinity, a � a0; a¤ects the time since which one can use the asymptotics
that is derived further. The probability density (113) has four parameters, c0; a, n and c1:
Usually, this is more than enough to �t the initial grain volume distribution near the point
a. Carr and Penrose (1999) showed that for function (113) the limit self-similar probability
density is the power function (85). It is essential that the powers n in the initial data (113)
and in the self-similar regime (85) are the same. In other words, parameter n is an invariant
of the evolution. It can be viewed as a "material constant" characterizing grain boundary
microstructure. Parameter n is responsible for inhomogeneity of the distribution of large
grains. As was mentioned, for n & 8 the self-similar distribution does not di¤er much fromthe exponential distribution. It is essential also that, as seen from (85), all other parameters
of the initial distribution, c0; a0; a, c1; are forgotten in the course of evolution. We rederive
Carr-Penrose result, taking also into account the leading terms that describe the evolution
to the self-similar regime.
In the vicinity of point a, functions G(u) and H(u); introduced in the previous Section,
have the form
G(u) =c0 (a� u)n+1
n+ 1
�1 + c1
n+ 1
n+ 2(a� u)
�; u 6 a;
H(u) =c0 (a� u)n+2
(n+ 1) (n+ 2)
�1 + c1
n+ 1
n+ 3(a� u)
�; u 6 a:
G(u) = H(u) = 0 for u > a: According to (99), for large � ; m (�) is determined from the
equationc0 (a�m)n+2
(n+ 1) (n+ 2)
�1 + c1
n+ 1
n+ 3(a�m)
�= e�� : (114)
For large � , e�� is small. Therefore, a�m is also small. Denote by � and y small functions
of � ;
� =
�1
c0(n+ 1) (n+ 2)
� 1n+2
e��=(n+2); y = a�m: (115)
The equation (114) takes the form
y
�1 + c1
n+ 1
n+ 3y
� 1n+2
= �: (116)
This equation determines y as a function of time parameter �: It is convenient to use further
y as a time parameter. The dependence of y on the real time is determined by (56), (88),
32
(115) and (116). Now we are going to �nd g(� ; u) as a function of y and u. To this end we
�rst obtain _m (�) � dm (�) =d� as a function of y. Since
G(m) =c0y
n+1
n+ 1
�1 +
n+ 1
n+ 2c1y
�; H(m) =
c0yn+2
(n+ 1) (n+ 2)
�1 +
n+ 1
n+ 3c1y
�;
from the last equation (112)
_m =H(m)
G(m)=
y
n+ 2
1 + n+1n+3
c1y
1 + n+1n+2
c1y: (117a)
Function �g in (110) depends, according to (113), on a � m � _mu: It is convenient to
introduce a new variable, w, u! w; as
w =_m
a�mu =
_m
yu: (118a)
Then,
a�m� _mu = (a�m)
�1� _m
a�mu
�= y (1� w) :
Since a�m� _mu takes values on [0; a] ; w and u changes on [0; 1] and [0; y= _m] ; respectively.
Function �g ( _mu+m) takes the form
�g ( _mu+m) = c0yn (1� w)n (1 + c1y (1� w)) :
From (110)
g (� ; u) = g1 (w)(1 + c1y (1� w))
1 + c1yn+1n+2
_m
y(119)
where
g1 (w) = (n+ 1) (1� w)n ; (120)
The factor _m=y is retained in (119) to achieve the invariance of the probability measure for
w�variable:g (� ; u) du = g1 (w)
1 + c1y (1� w)
1 + c1yn+1n+2
dw: (121)
Functions g1 (w) has the meaning of probability density for y = 0:
1Z0
g1 (w) dw = 1: (122)
33
When � ! 1; y tends to zero, _m=y tends to 1= (n+ 2) ; and probability density
g1 (w) = (n+ 2) ; written in terms of the relative grain volume u, coincides with steady-state
distribution (85) as claimed.
The range of validity of formulas obtained is determined by the condition that all points
�u = _mu+m; 0 6 u 6 y= _m = (a�m) = _m are in the segment [a0; a]. Functions y and _m tend
to zero as � !1; therefore these points are in [a0; a] as soon as
a0 6 m (�) :
Denote by � 0 the instant when this occurs: a0 = m (� 0) : Function _m (�) is positive as follows
from (117a). Therefore, m (�) is a growing function for � > � 0; and the relations obtained
hold for all � > � 0: The smaller a0 the sooner the relations derived can be applied.
VIII. MICROSTRUCTURE ENTROPY IN SELF-SIMILAR GRAIN GROWTH
In this Section we justify the two statements made in Introduction in case of self-similar
grain growth: total microstructure entropy of the system, Sm jV j = Sm; decays while entropyper one grain, S� = Sm=N; grows. First of all, we have to de�ne microstructure entropy.
In (Berdichevsky 2005, 2008) it was identi�ed with logarithm of phase volume. This as-
sumed implicitly that the number of degrees of freedom does not change. Besides, energy
surfaces should remain at least approximately energy surfaces in the process of evolution.
In grain size/grain volume dynamics the number of degrees of freedom does change, and
microstructure entropy needs to be rede�ned. To motivate the new de�nition, consider the
grain volume dynamics in the self-similar regime. We take for the grain volume probability
density in unbounded polycrystals the function,
fss (t; v) =1
�v(t)e�v=�v(t): (123)
It coincides with (8) and can be obtained also from the assumption that the grain microstruc-
ture is the most chaotic, or, in more precise terms, that all grain volumes, obeying to the
constraint (43), are equiprobable (Berdichevsky 2011). Logarithm of such phase volume �
for a large number of grains N is
ln � = N (ln �v + 1) : (124)
34
To avoid logarithms of dimensional quantities, we introduce some characteristic volume v0
and write (124) as
ln�
vN0= N
�ln�v
v0+ 1
�: (125)
By v0 one can mean, for example, the smallest observable grain volume.
Let us try now for entropy of grain structure the Gibbs-Shannon formula
S� = �Zfss (t; v) ln (fss (t; v) v0) dv: (126)
For probability distribution (123),
S� = ln�v
v0+ 1: (127)
Hence, if we introduce total microstructure entropy of the system for the self-similar growth
as
Sm = NS�; (128)
where S� is given by (126), then Sm will coincide with the phase volume in the self-similar
regime. The choice of v0 �xes the value of an additive constant in entropy.
As follows from (127), S� grows in the self-similar regime, because the average grain size
grows. The behavior of total microstructure entropy Sm depends on the decay rate of N
and the growth rate of S�: In the self-similar regime N decays faster than S� grows. Indeed,
di¤erentiating (128) with respect to � ; we have
dSmd�
=dN
d�
�ln�v
v0+ 1
�+N
�v
d�v
d�:
Replacing here dN=d� by its expression from (71) we obtain
dSmd�
= �N�v
d�v
d�ln�v
v0: (129)
Hence, Sm decays as claimed. Apparently, the same occurs near the self-similar regime due
to smallness of deviations. Emphasize that equation (127) holds only if the grain volume
distribution is exponential. If it is not, for example, it is the power distribution (85), then
S�, computed by (126), di¤ers from (127) and depends on n,
S� = ln�v
v0+ b0(n); b0(n) =
n
n+ 1� ln n+ 1
n+ 2: (130)
35
Function b0(n) is a growing function of n, tending to unity as n!1: Accordingly, entropy
(130) is smaller than entropy (127), as it must be, because (127) corresponds to most chaotic
microstructures with the largest possible value of entropy. For the grain volume distribution
(85), entropy per one grain, S�; grows in the self-similar regime. The total microstructure
entropy decays,dSmd�
= �N�v
d�v
d�
�ln�v
v0� (1� b0 (n))
�; (131)
as long as ln (�v=v0) > 1 � b0 (n) : The di¤erence 1 � b0 (n) does not exceed 1 � b0 (0) � 0:3;and the right hand side of (131) is negative unless the average grain volume is extremely
small. However, even in this case, after some transitional period, the right hand side of (131)
becomes negative, and the total microstructure entropy decays.
IX. MICROSTRUCTURE ENTROPY EVOLUTION
For an arbitrary process entropy is still to be de�ned. A natural choice is
S� = �Zf (t; v) ln (f (t; v) v0) dv: (132)
This is the choice which we will use. For the self-similar regime (132) transforms to (126).
To derive the evolution equation for S� we write it as
S� = ln�v
v0+ S0; S0 = �
Zg (� ; u) ln g (� ; u) du; (133)
and use equation for g (� ; u) (89), which we put in the form
@ ln g
@�+ (u (1� q)� 1) @ ln g
@u= 2q � 1: (134)
Multiplying (134) by g, summing with equation (8) multiplied by ln g and integration over
u, we havedS0d�
= qS0 + 1� 2q + g ln g
����u=0
:
Invoking now (94) we obtain
dS0d�
= qS0 + 1� 2q + q ln q: (135)
Since S0 can take any negative value by a corresponding choice of g(0; u) (see Appendix A),
for any choice of q(0) = g(0; 0); the right hand side of (135) can be made negative at initial
36
instant. Therefore, initially S0 may decay. However, after some transitional period, the
system approaches the self-similar regime, and, at least in a close vicinity of the self-similar
regime, S0 grows.
The same argument is applied to S� and Sm = S= jV j = S�=�v : from (135), (133), (128)
and (90)dS�
d�= q
�S� � ln �v
v0
�+ 1� q + q ln q (136)
�vdSmd�
= �q ln �vv0+ 1� q + q ln q: (137)
For example, in (137), for any initial value of ln (�v=v0) ; one can always choose initial data
for q = g(0; 0) in such a way that the right hand side of (137) is positive initially. However,
if we impose on the initial data a physically non-constraining condition,
�g(u)H(u)
G(u)26 const; (138)
then q(�) is bounded from above, and, due to the growth of ln (�v=v0) ; after a possible
transitional period, Sm decays.
X. CONSTITUTIVE EQUATION FOR MICROSTRUCTURE ENTROPY
As follows from the relations obtained, the probability density keeps memory of initial
conditions when the system approaches the self-similar regime, and so do energy and entropy.
Let us show that for � > � 0 entropy, energy and �v are linked by an equation of state
S� = ln�v
v0+ S0(X;n); (139)
where X is a dimensionless energy,
X =E
2� N�
�v4�=3
�2=3 = 2
3
Um
��v
4�=3
�1=3=
y= _mZ0
'(u)g (� ; u) du: (140)
Equation of state holds independently on the initial data parameters, a0; a; c0; c1: The only
parameter of microstructure that enters (139), is the parameter n, but, as was mentioned, n
is an invariant of the evolution process.
37
Indeed, from (119) and (140),
X =
1Z0
'(u)g1 (w)1 + c1y (1� w)
1 + c1yn+1n+2
dw; u =y
_mw = w (n+ 2)
1 + n+1n+2
c1y
1 + n+1n+3
c1y: (141)
We see what X is a function of c1y and n:
X = X(c1y; n): (142)
The variables c1y and n consumes all the dependence of X on the initial data.
Let us �nd now entropy of microstructure. We have
S� = �Zf (� ; v) ln (f (� ; v) v0) dv = ln
�v
v0+ S0; S0 = �
Zg ln gdu:
Plugging here (119) we get
S0 = �1Z0
g1 (w)(1 + c1y (1� w))
1 + c1yn+1n+2
ln
"g1 (w)
(1 + c1y (1� w)) 1 + c1yn+1n+3�
1 + c1yn+1n+2
�2#dw: (143)
So, S0 is also a function of c1y and n. Eliminating parameter c1y we obtain the equation of
state (139).
Let us get the �rst terms of the corresponding Taylor series. As � !1; X takes the value
X0 =n+ 1
n+ 2
n+2Z0
' (x)
�1� x
n+ 2
�ndx:
This value is practically independent on n and for n = 1 is equal to 0.88. As � ! 1; S0
tends to the value b0(n) (130). This value is also practically constant with b0(1) = 1: The�rst terms of Taylor expansion for integrals (141) and (143) are
X = X0 +c1y
n+ 2
n+ 1
n+ 2
n+2Z0
�' (x) (1� x) +
n+ 1
n+ 3'0 (x)x
��1� x
n+ 2
�ndx
S0 = b0(n) +c1y
n+ 2
2
(n+ 2) (n+ 3):
Hence,
S0 = b0(n) + b1(n) (X �X0) ; (144)
38
FIG. 7: The dependence of the coe¢ cient b1 in (144) on n.
where
b1(n) =2
(n+ 1) (n+ 3)
,n+2Z0
�' (x) (1� x) +
n+ 1
n+ 3'0 (x)x
�dx:
The dependence of the coe¢ cient b1 on n is shown in Fig. 7. Note that b1 < 0:
Temperature of microstructure, Tm; de�ned by usual thermodynamics relation,
1
Tm=@Sm@Um
; Sm =1
�vS�
is
Tm =2�
b1
��v
4�=3
�2=3:
The temperature is negative. As was mentioned in (Berdichevsky 2008), for microstructures
this is not as exotic as it is in classical thermodynamics; negativeness of microstructure tem-
perature indicates that growth of chaos is accompanied by decay of energy. The dependence
of entropy on dimensionless energy in the vicinity of the self-similar regime is shown in Fig.
8. In this Figure the thick line corresponds to the self-similar evolution while thin lines
show the dependence of S0 on X for di¤erent n. Along each line S0 grows and X decreases
approaching the values in the self-similar regime. The negative slope of the lines indicates
negativeness of temperature in the process of grain growth..
39
FIG. 8: Equation of state: dependence of microstructure entropy on dimensionless energy X and
inhomogeneity parameter n.
XI. EXAMPLES OF GRAIN STRUCTURE EVOLUTION
In this Section several examples of grain structure evolution are presented. We assume
that the initial grain volume probability density can be satisfactorily represented by the
power law
�g(u) =
8<: c0 (a� u)n (1 + c1 (a� u)) 0 6 u 6 a
0 u > a:
The constraints (83) determine uniquely the constants c0 and c1. Then on [0; a]
�g(u) = (n+ 2) (a� u)n (a (2a� (n+ 3)) + (n+ 3) (n+ 2� a)u) =an+3: (145)
The probability density contains two parameters, a and n. Positiveness of �g(u) restricts, as
easy to see, the range of admissible a and n:
n+ 3
26 a 6 n+ 3: (146)
Besides, we assume, that n > 0 to remain within the usually observed distributions.Grain volume probability densities for a = 7, n = 11 and three times � = 0; � = 1 and
� = 10 are shown in Fig. 9. Probability density for � = 10 practically does not change
40
FIG. 9: Evolution of relative grain volume probability density for a = 7, n = 11.
FIG. 10: The dependence of squared average grain size on dimensionless time. Linear part of the
curve corresponds to the self-similar grain growth.
further, i.e. it is the �nal self-similar distribution. The corresponding dependence of squared
average grain size (v=�v (0))�2=3 as a function of dimensionless time x = t�v (0)�2=3 ; where t
is de�ned by (56), is shown in Fig. 10; the horizontal coordinate ispx. Linear part of the
curve corresponds to self-similar grain growth according to (??). For numerical values (52),
x = 0:08 M tphys�v (0)�2=3 :
41
FIG. 11: The dependence of squared average grain size on time for two di¤erent grain size distrib-
utions.
For M � 10�2mm2=s; which approximately corresponds to data for pure iron at 850C�
(Vandermeer et al. (1994))5 and for �v (0)�1=3 = 50�m; x � 0:4 tphys; tphys being measured
in seconds. As is seen from Fig. 10, the time of approaching the self-similar regime is very
fast, about 2-3 s. During this time the average grain size increases for about 30%. At lower
temperature T=550C�; M � 10�6mm2=s: Accordingly x � 4 � 10�4 tphys; and reaching theself-similar regime takes about 5000 s. It is interesting that despite quite drastic change
in probability density, dimensionless energy changes a little, a few per cents, while entropy
S0 changes noticeably. In this regard thermodynamics of grain growth is reminiscent of
"entropic elasticity" of rubbers and polymers, when energy is practically constant in isother-
mic processes, and the most sensitive thermodynamic function is entropy. Of course, such
an analogy is super�cial, because in grain growth dimensionless energy is subject to small
changes, while dimensional energy does change.
It is important to notice that the time of reaching the self-similar regime depends on the
initial grain volume distribution. This is illustrated by Fig. 11, showing the growth of average
size as a function of time for n = 20 and two values of a, a = 11:5, and a = 22:9.
5 In these estimates, the constant 0.08M is identi�ed with the rate constant K in Vandermeer et al. (1994).
42
FIG. 12: Evolution of relative grain volume probability density for a = 2, n = 1.
FIG. 13: Evolution of relative grain volume probability density for a = 7, n = 7.
Evolution of the grain volume probability density for another set of parameters, a = 2,
n = 1 is shown in Fig. 12. Here the values of parameters are intentionally chosen at the
border of region (146) to observe sharp changes. Otherwise, the changes are not signi�cant
as Fig. 13 shows for a = 7, n = 7.
The change in time of energy and entropy is depicted in Fig. 14 for a = 11:5, n = 20.
It is seen that dimensionless energy changes insigni�cantly, while entropy change is more
43
FIG. 14: Time dependence of S0 and X for a = 11:5, n = 20.
pronounced.
XII. CONCLUDING REMARKS
The model of grain growth presented in the paper exhibits the following features:
� Probability density of grain volumes, f(t; v); converges to a self-similar distribution,
f1(t; v) =1
�v(t)g1 (u) ; u =
v
�v(t)
�v(t) being the averaged grain volume, if the initial distribution for the largest grains
has the form
g(0; u) =
8<: c0 (a� u)n (1 + c1 (a� u)) ; a0 6 u 6 a
0 u > a: (147)
Formula (147) does not seem physically constraining and contains enough number of
parameters to �t experimental data.
� The limit self-similar distribution is
g1 (u) =
8<: n+1(n+2)n+1
(n+ 2� u)n 0 6 u 6 n+ 2
0 u > n+ 2:
44
It does not depend on the parameters of the initial distribution c0; a, c1; and inherits
only the power n. The power n characterizes the inhomogeneity of the distribution of
large grains.
� For large n (it is enough that n > 8), the power distribution is practically coincided
with exponential distribution,
g1 (u) = e�u; 0 6 u <1;
and approximate well exponential data.
� Energy of grain boundaries per unit mass, Um; depends on microstructure. The dimen-sionless energy of grain boundaries,
X =2
3
Um
��v
4�=3
�1=3;
being grain boundary energy per unit area, evolves in the course of grain growth.
It�s limit value at the self-similar regime is practically independent on the initial grain
volume distribution and equal to 0.88. The initial values of X are within the limits
(see Appendix A)' (a)
a6 X 6 ' (1)
where ' (a) is the function (51). The ratio ' (a) =a decays with increasing a to zero;
' (1) = 1: Therefore, for su¢ ciently large a, the range of change of X can be practically
[0; 1] :
� Entropy of grain boundary microstructure per one grain, S�; is de�ned as
S� = �Zf(t; v) ln (f(t; v)v0) dv; (148)
where v0 is some characteristic grain volume, e.g. the minimum observable grain vol-
ume. Accordingly, entropy of microstructure per unit volume, Sm , is
Sm =1
�vS�:
Formula (148) allows one to measure entropy of microstructure experimentally.
45
� In the self-similar grain growth,
Sm =1
�v
�ln�v
v0+ 1
�; S� = ln
�v
v0+ 1:
Therefore, in the self-similar grain growth total entropy, Sm� (volume of polycrystal),decays, while entropy per one grain, S�; grows.
� There is an equation of state linking energy of microstructure Um; entropy of mi-crostructure Sm; averaged grain volume �v; and the characteristic of inhomogeneity of
large grain distribution n:
�vSm = ln�v
v0+ S0 (X;n) :
The dependence S0 on X and n in the vicinity of the self-similar growth is shown in
Fig.8.
It would be interesting to see to which extent these consequences of a mathemat-
ical model are supported by experimental data. If they are, then the well-developed
thermodynamic technique can be employed to obtain models of grain growth taking
into account multiple complications characteristic to this process: e¤ect of precipitates
and impurities, dislocation networks, plasticity, etc.
XIII. APPENDIX A. ENERGY AND ENTROPY AS FUNCTIONALS OF GRAIN
VOLUME DISTRIBUTION
Dimensionless energy of grain boundary structure,
X =
aZ0
' (u) g (u) du; (149)
can be considered as a linear functional of relative volume probability density g (u) : Function
g (u) is an arbitrary function obeying the constraints
g > 0;aZ0
g (u) du = 1;
aZ0
ug (u) du = 1: (150)
In this Appendix we show that possible values of X are within the limits
1
a'(a) 6 X 6 '(1): (151)
46
Moreover, we show that the lower bound is achieved on the minimizing sequence
g(u) = p�(u� ") + (1� p) � (u� (a� ")) ; "! 0 (152)
while the upper bound is reached for
g(u) = � (u� 1) : (153)
In (152) the limit values of probabilities p and 1� p are
p =a� 1a
; 1� p =1
a: (154)
It is assumed in (151)-(153) that function ' (u) possesses the same features as function (51):
' (u) > 0; ' (0) = 0; '0 (0) > 0; '00 (0) < 0: The length a of the support of g(u)
must be not less than 1 as follows from (150):
1 =
aZ0
ug (u) du 6 a
aZ0
g (u) du = a:
Formulas (152) and (153) are quite interesting. (152) shows that bimodal grain boundary
structures may have smaller energy than the grain structures with monotonous grain volume
distribution. To obtain the bimodal distribution with the smallest energy for the case when
relative grain volumes change on the segment [0; a], one has to take the smallest grains with
concentration (a � 1)=a and largest grains with concentration 1=a. According to (153), thehomogeneous distribution of grain volumes (all grains have the same volume) correspond
to maximum possible energy. The fact that the evolution does not go to create bimodal
distributions is caused by dissipative and "entropic" e¤ects generating chaotization.
The proof of (151)-(153) proceeds as follows. Consider the variational problem
Xmin = ming2(150)
aZ0
' (u) g (u) du: (155)
Notationming2(150) means minimization over all functions g(u) obeying the constraints (150).
Let us rewrite (155) as a minimax problem6,
Xmin = min
g>0
Zgdu=1
max�
24 aZ0
' (u) g (u) du+ �
0@ aZ0
ug (u) du� �
1A35 :6 Regarding this trick and further inequality (156) see, e.g., Berdichevsky 2009, Sect. 5.6 and 5.10.
47
Since min max > maxmin;
Xmin > max�
min
g>0;
Zgdu=1
24 aZ0
(' (u) + �u) g(u)du� �
35 : (156)
Minimization over g(u) is conducted explicitly: obviously, one has to concentrate g(u) at the
point of minimum of '(u) + �u: We get
Xmin > max�
hminu('(u) + �u)� �
i: (157)
Due to the above-mentioned features of '(u)
� (�) = minu('(u) + �u) =
8<: 0 � > �'(a)a
' (a) + �a � 6 �'(a)a
:
Function � (�)� � has the only maximum at the point � = �' (a) =a: Hence
' (a)
a6 Xmin: (158)
Let us obtain now the upper estimate of Xmin: Consider the functional X (g) on the function
g(u) (152). It becomes a function of two parameters, p and " :
X(g) = p' (") + (1� p)' (a� ") :
Parameters p and " are linked by the last constraints (150)
p"+ (1� p) (a� ") = 1:
Thus,
p =a� "� 1a� 2" :
Hence
Xmin 6 X(g) =a� "� 1a� 2" ' (") +
1� "
a� 2"' (a� ") :
For "! 0; ' (")! 0; ' (a� ")! ' (a) and
Xmin 6' (a)
a: (159)
From (158) and (159) Xmin = ' (a) =a:
48
The upper bound is derived similarly:
Xmax = maxg2(150)
aZ0
' (u) g (u) du
= maxg>0;
Rgdu=1
min�
24 aZ0
' (u) g (u) du+ �
0@ aZ0
ug (u) du� 1
1A356 min
�max
g>0;Rgdu=1
24 aZ0
(' (u) + �u) g (u) du� �
35= min
�
hmaxu(' (u) + �u)� �
i: (160)
The right hand side of (160) can be increased by taking any trial value �. Let us take
� = �'0 (1) : Since '00 (u) < 0; '0 (u) decreases monotonously, and maximum of ' (u)�'0 (1)ucan be achieved only at one point u = 1. Hence,
Xmax 6 maxu(' (u) + �u)� �
����=�'0(1)
= ' (1) : (161)
On the other hand on an admissible trial function g(u) = �(u� 1);
X(g) = ' (1) 6 Xmax: (162)
The upper bound (151) follows from (161) and (162).
Entropy functional,
S0(g) = �aZ0
g (u) ln g (u) du;
in contrast to energy functional is not bounded from below. It is enough to concentrate g(u)
in the vicinity of the point u = 1 to send S0 to �1:
XIV. APPENDIX B. GRAIN VOLUME DYNAMICS IN FINITE POLYCRYSTALS
Continuum theory of grain size dynamics and grain volume dynamics described in Sections
3 and 4 formally makes sense for polycrystals of �nite volume with any smooth initial grain
size/grain volume distributions. However, for �nite volumes, it is intrinsically inconsistent.
In dynamics of a �nite number of grains, the evolution ends when only one grain survives
49
FIG. 15: Evolution of the support of grain volume density.
and occupies the entire polycrystal volume. In continuum theory the �nal stage of evolution
is a self-similar grain growth even in a �nite crystal. First, we elucidate this point, and then
show how to modify the setting of the problem to get rid of such drawback.
The maximum possible value of grain volumes is jV j : Accordingly, the maximum possiblevalues of relative grain volume is jV j =�v = N(�): Consider the grain volume evolution with the
initial distribution of relative grain volumes�g(u):We assume that�g(u) > 0 for 0 6 u 6�umax;and �g(u) = 0 for �umax < u < N(0): For � > 0 g (� ; u) is non-zero for 0 6 u 6 umax (�) ;
and zero for umax (�) < u 6 N(�): We expect that at the �nal state, when the �nal grain
occupies the entire polycrystal, N = 1, �v = jV j ; umax = 1: So, the evolution of the supportof g (� ; u) should look like the one shown in Fig. 15, where � f is the �nal time when all grain
boundaries disappear. Function g (� f ; u) is to be � (u� 1) : In fact, within the framework oftheories of Sections 3, 4, the evolution may look di¤erently. For example, for exact solution
of Section 11, function umax (�) approaches the asymptotic value n+ 2; which is larger then
unity while g(� ; u) di¤ers from ��function for all � : Besides, this solution is meaningless assoon as V=�v(�) = N (�) becomes smaller than unity.
The continuum theory can be �xed in the following way. Let the grain ensemble has the
only grain with the maximum volume, vmax: We set the initial density of the ensemble to
be�� (v) + � (v � vmax) ; where�� (v) is a smooth function on [0; vmax] : Further, by � (t; v) we
mean the grain density which has the initial value�� (v) : Then the di¤erential equation (42)
50
does not change, and the only changes occur in the integral relations:
vmax(t)Z0
� (t; v) dv = N � 1; (163)
vmax(t)Z0
v� (t; v) dv + vmax (t) = jV j ; (164)
�v (t) =jV jN=
vmax(t)R0
v� (t; v) dv + vmax (t)
vmax(t)R0
� (t; v) dv + 1
: (165)
Function vmax (t) is determined by the law of evolution of grain volume (59)
dvmax (t)
dt=
1
�v2=3 (t)(vmax (t)� �v) ; vmax (0) = vmax: (166)
The evolution stops at the instant tf when N = 1. Then from (163)
vmax(tf)Z0
� (tf ; v) dv = 0;
and, since � > 0;� (tf ; v) = 0:
From (164)we obtain
vmax (tf ) = jV j ;
as it should be, and from (165)
�v (tf ) = vmax (tf ) :
These relations, rewritten in terms of function g = ��v2= jV j yield the same equation forg (t; u) (89) and modi�ed integral relations
umax(�)Z0
gdu = 1� 1
N (�);
umax(�)Z0
ugdu = 1� umax (�)
N (�): (167)
Functionsm (�) ; M (�) ; G(u); H(u) and p (�) are introduced by the same relations as before.
However, due to (167), equation for m (�) is replaced by the equation
H (m (�)) + (1�G (0)) (umax �m (�)) = e�� : (168)
51
As easy to see
1�G (0) =1
N(0):
For large N(0), 1�G (0) is small. Therefore, the additional term in (168) does not a¤ect theinitial stage of the evolution of a large number of grains.
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